Insights into the emergence of mass from studies of pion and kaon structure

doi:10.1016/j.ppnp.2021.103883

Progress in Particle and Nuclear Physics

Volume 120, September 2021, 103883

https://doi.org/10.1016/j.ppnp.2021.103883 Get rights and content

Abstract

There are two mass generating mechanisms in the standard model of particle physics (SM). One is related to the Higgs boson and fairly well understood. The other is embedded in quantum chromodynamics (QCD), the SM’s strong interaction piece; and although responsible for emergence of the roughly 1 GeV mass scale that characterises the proton and hence all observable matter, the source and impacts of this emergent hadronic mass (EHM) remain puzzling. As bound states seeded by a valence-quark and -antiquark, pseudoscalar mesons present a simpler problem in quantum field theory than that associated with the nucleon. Consequently, there is a large array of robust predictions for pion and kaon properties whose empirical validation will provide a clear window onto many effects of both mass generating mechanisms and the constructive interference between them. This has now become significant because new-era experimental facilities, in operation, construction, or planning, are capable of conducting such tests and thereby contributing greatly to resolving the puzzles of EHM. These aspects of experiment, phenomenology, and theory, along with contemporary successes and challenges, are reviewed herein. In addition to providing an overview of the experimental status, we focus on recent progress made using continuum Schwinger function methods and lattice-regularised QCD. Advances made using other theoretical tools are also sketched. Our primary goal is to highlight the potential gains that can accrue from a coherent effort aimed at finally reaching an understanding of the character and structure of Nature’s Nambu–Goldstone modes.

Section snippets

Emergence of mass

When considering the origin of mass in the standard model of particle physics (SM), thoughts typically turn to the Higgs boson because couplings to the Higgs are responsible for every mass-scale that appears in the SM Lagrangian. The notion behind this Higgs mechanism for mass generation was introduced more than fifty years ago [1], [2], [3] and it became an essential piece of the SM. In the ensuing years, all the particles in the SM Lagrangian were found; although the Higgs boson proved

Gluon mass scale

The QCD trace anomaly exerts a material influence on every one of QCD’s Schwinger functions; but for those unfamiliar with analyses of QCD’s gauge sector, the most striking impact, perhaps, is that expressed in the gluon two-point function. Interaction induced dressing of a gauge boson is expressed through the appearance of a nonzero polarisation tensor, $Π_{μ ν} (k)$ . The generalisation of gauge symmetry to the quantised theory is expressed in Slavnov–Taylor identities (STIs) [53], [54]. Regarding

Essentials of light-front wave functions

If one seeks to describe a given hadron’s measurable properties in terms of the probabilities typical of quantum mechanics, then the hadron’s LFWF, $ψ_{H} (x, {\vec{k}}_{⊥}; P)$ , takes a leading role. Here [158], [159]: $P$ is the total four-momentum of the system, $x$ is the light-front longitudinal fraction of this momentum, and ${\vec{k}}_{⊥}$ is the light-front perpendicular component of $P$ . In principle, this LFWF is an eigenfunction of a QCD Hamiltonian defined at fixed light-front time and may be obtained by

Electromagnetic transition form factors

There are few rigorous QCD predictions for processes that involve strong dynamics, like hadron elastic and transition form factors. The cleanest are linked to $γ^{*} γ^{(*)} \to Π$ transition form factors, $G_{Π} (Q^{2})$ , where $Π$ is a charge-neutral pseudoscalar meson and $Q$ is the virtual photon momentum. With the second photon being real and isolating a given $q \bar{q}$ component of $Π$ , then there exists $Q_{0} > Λ_{QCD}$ such that [167] $Q^{2} G_{Π}^{q} (Q^{2}) \overset{Q^{2} > Q_{0}^{2}}{\approx} 4 π^{2} f_{Π}^{q} e_{q}^{2} w_{Π}^{q} (Q^{2}),$ where: $f_{Π}^{q}$ is the pseudovector projection of the $q \bar{q}$ piece of

Forward compton scattering amplitude

A description of the status of experiment and theory for pion and kaon distribution functions as it was ten years ago may be found in Ref. [52]. An update on experiment is presented in Section 9.3. Here it is worth recapitulating a few facts in order to establish the context for that update and recent developments in theory. Thus, the hadronic tensor relevant to inclusive deep inelastic lepton+pseudoscalar-meson ( $ℓ Π$ ) scattering can be expressed via two invariant structure functions [256]. With

Kaon distribution functions

So far as kaon DFs are concerned, the only information available is a forty year old DY measurement of the $K^{-} ∕ π^{-}$ structure function ratio [340], just eight points of data; and the past decade has seen a raft of model and theory calculations compared with the data on $u^{K} (x; ζ_{5}) ∕ u^{π} (x; ζ_{5})$ inferred therefrom, e.g. Refs. [103], [104], [169], [173], [174], [271], [303], [304], [308], [341], [342], [343]. Mathematically, Eqs. (5.77)–(5.80) ensure that the large- $x$ power-law exponents of $u^{π} (x, ζ)$ and $u^{K} (x, ζ$

Generalised transverse-momentum dependent parton distribution functions

As highlighted already, experiment, phenomenology and theory have long focused on drawing one dimensional (1D) images of hadrons. This effort continues because many puzzles and controversies are unresolved, e.g. the large- $x$ behaviour of meson structure functions and the glue and sea content of NG modes. Yet, notwithstanding the need for new, precise data on 1D distributions and related predictions with a traceable connection to QCD, the attraction of generalised parton distributions (GPDs) and

Formulation

Discretising QCD on a Euclidean lattice, originally proposed more than forty years ago [376], provides a first-principles means of solving QCD in the strong-coupling regime. The starting point for a lattice computation is the discretised, Euclidean path integral, wherein physical “observables” $O$ are computed as $〈 O 〉 = \frac{1}{Z} \prod_{n, μ} d U_{μ} (n) \prod_{n} d ψ (n) \prod_{n} d \bar{ψ} (n) O (U, ψ, \bar{ψ}) e^{- (S_{G} [U] + S_{F} [U, ψ, \bar{ψ}])},$ where $U_{μ} (n)$ are 3 × 3 unitary matrices representing the gauge fields, $ψ, \bar{ψ}$ are Grassmann variables representing the fermion

Sullivan process

In specific kinematic regions, the observation of recoil nucleons (N) or hyperons (Y) in the semi-inclusive measurement $e p \to e^{'} (N or Y) X$ can reveal features associated with correlated quark–antiquark pairs in the nucleon, referred to as the “meson cloud” or “five-quark component” of the nucleon. At low values of $t$ , the four-momentum transfer from the initial proton to the final nucleon or hyperon, the cross-section displays behaviour characteristic of meson pole dominance. Illustrated in Fig. 9.40,

Measurements on the horizon

The broad international science programme aimed at understanding pion and kaon structure and the SM mechanisms behind the emergence of hadron masses requires a strong, constructive interplay between experiment, phenomenology and theory. Experimental prospects must be matched and guided by new theoretical insights, assisted by rapid advances in computing and high-level QCD phenomenology. The identification and conduct of those experiments which can best lead to novel theory insights and

Epilogue

Existence of the pion was predicted eighty-five years ago [536]; yet, even now, very little is known about its structure. The pion’s mass is measured with precision — to one-part in a million; but its radius is only constrained to within 1% and even this is debatable, given the uncertainties now attendant upon radius measurements made using electron+hadron scattering experiments [537]. Regarding the kaon, discovered over seventy years ago [117] and the first known particle to possess

Abbreviations

The following abbreviations are used in this manuscript:

2PI	two particle irreducible
BS (BSE)	Bethe–Salpeter (equation)
CEA	Cambridge (Massachusetts) electron accelerator
CERN	European Laboratory for Particle Physics
CLAS	detector in Hall-B at JLab
DA	distribution amplitude
DCSB	dynamical chiral symmetry breaking
DESY	Deutsches Elektronen-Synchrotron (accelerator in Hamburg)
DF	distribution function
DSE	Dyson–Schwinger equation
DVCS	deeply virtual Compton scattering
DVMP	deeply virtual meson production
DY	Drell–Yan

Acknowledgements

We are grateful for assistance and insightful comments from D. Binosi, S. J. Brodsky, K.-L. Cai, Z.-F. Cui, O. Denisov, R. Ent, T. Frederico, J. Friedrich, J. Karpie, N. Karthik, C. Mezrag, W.-D. Nowak, S. Platchkov, J. Qiu, C. Quintans, J. Rodríguez-Quintero, G. Salmè, J. Segovia, R. Sufian, S.-S. Xu, J.-L. Zhang. CDR is supported in part by the Jiangsu Province Hundred Talents Plan for Professionals, China; TH and DGR by the U.S. Department of Energy, Office of Science, Office of Nuclear

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Progress in Particle and Nuclear Physics

ReviewInsights into the emergence of mass from studies of pion and kaon structure

Abstract

Section snippets

Emergence of mass

Gluon mass scale

Essentials of light-front wave functions

Electromagnetic transition form factors

Forward compton scattering amplitude

Kaon distribution functions

Generalised transverse-momentum dependent parton distribution functions

Formulation

Sullivan process

Measurements on the horizon

Epilogue

Abbreviations

Acknowledgements

Phys. Lett.

Phys. Rep.

Nuclear Phys. B

Nucl. Phys. B Proc. Suppl.

Nuclear Phys. B

Nuclear Phys. B

Nuclear Phys. B

Prog. Part. Nucl. Phys.

Nuclear Phys. B

Phys. Lett. B

Phys. Rep.

Prog. Part. Nucl. Phys.

Nuclear Phys. B

Phys. Rep.

Nuclear Phys. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Lett. B

Phys. Lett. B

Rev. Modern Phys.

Rev. Modern Phys.

Prog. Theor. Exp. Phys.

Ann. Rev. Nucl. Part. Sci.

Eur. Phys. J. A

Quantum Field Theory

Phys. Rev. D

Phys. Rev. D

Proc. Natl. Acad. Sci.

Proc. Natl. Acad. Sci.

Proc. Natl. Acad. Sci.

Phys. Rev. D

Few Body Syst.

J. Phys. G

Eur. Phys. J. C

Phys. Rev.

Nuovo Cim.

Phys. Rev.

J. Phys. G.

Phys. Rev. D

Phys. Rev. C

Review
Insights into the emergence of mass from studies of pion and kaon structure